Christoph Koutschan - cp's OEIS frontend

A381554 Number of 2*n X 8 binary arrays with row sums 4 and column sums n, avoiding the patterns 010 and 101 in any row and column.

1, 18, 324, 2776, 34586, 575270, 10061200, 194929482, 4115573632, 91947761368, 2167131637540, 53550929019486, 1376887129235964, 36670419524921146, 1007581514656491404, 28454294028011307236, 823234343053953729538

Offset: 0

Christoph Koutschan, Feb 27 2025

nonn

			A few solutions for n = 3 are:
  0 0 0 0 1 1 1 1   0 0 0 1 1 1 1 0   0 1 1 0 0 1 1 0   1 0 0 1 1 1 0 0
  0 0 0 0 1 1 1 1   0 0 0 1 1 1 1 0   0 1 1 1 0 0 0 1   1 1 0 0 1 1 0 0
  0 0 0 0 1 1 1 1   1 0 0 1 1 0 0 1   0 1 1 1 0 0 0 1   1 1 0 0 0 1 1 0
  1 1 1 1 0 0 0 0   1 1 1 0 0 0 0 1   1 0 0 1 1 0 0 1   0 1 1 0 0 0 1 1
  1 1 1 1 0 0 0 0   1 1 1 0 0 0 0 1   1 0 0 0 1 1 1 0   0 0 1 1 0 0 1 1
  1 1 1 1 0 0 0 0   0 1 1 0 0 1 1 0   1 0 0 0 1 1 1 0   0 0 1 1 1 0 0 1

Cf. A381551, A381553.

A381553 Number of 2*n X 6 binary arrays with row sums 3 and column sums n, avoiding the patterns 010 and 101 in any row and column.

Original entry on oeis.org

1, 8, 64, 368, 2776, 25880, 251704, 2629080, 28964248, 331032312, 3907675376, 47392320240, 587548108400, 7421689479608, 95248568409312, 1239031994818680, 16306127772957216, 216760982171930144, 2906731043068293952, 39278080769921432856, 534346120254755625336

Offset: 0

Christoph Koutschan, Feb 27 2025

nonn

			A few solutions for n = 3 are:
  0 0 0 1 1 1   0 0 0 1 1 1   0 0 0 1 1 1   0 0 0 1 1 1   0 0 0 1 1 1
  0 0 0 1 1 1   0 0 0 1 1 1   0 0 0 1 1 1   0 0 0 1 1 1   0 0 0 1 1 1
  0 0 0 1 1 1   0 0 1 1 1 0   0 1 1 0 0 1   0 1 1 1 0 0   1 0 0 0 1 1
  1 1 1 0 0 0   1 1 1 0 0 0   1 1 1 0 0 0   1 1 1 0 0 0   1 1 1 0 0 0
  1 1 1 0 0 0   1 1 1 0 0 0   1 1 1 0 0 0   1 1 1 0 0 0   1 1 1 0 0 0
  1 1 1 0 0 0   1 1 0 0 0 1   1 0 0 1 1 0   1 0 0 0 1 1   0 1 1 1 0 0

Christoph Koutschan, Table of n, a(n) for n = 0..70
Robert Dougherty-Bliss, Christoph Koutschan, Natalya Ter-Saakov, and Doron Zeilberger, The (Symbolic and Numeric) Computational Challenges of Counting 0-1 Balanced Matrices, Enumerative Combinatorics and Applications 5:2, Article #S2R14, 2025.

Cf. A381551, A381554.

A381551 Number of 2*n X 4 binary arrays with row sums 2 and column sums n, avoiding the patterns 010 and 101 in any row and column.

Original entry on oeis.org

1, 4, 16, 64, 324, 1764, 10000, 58564, 350464, 2131600, 13133376, 81757764, 513294336, 3245580900, 20646241344, 132021769104, 848031024996, 5468890936356, 35392361925904, 229761144199876, 1495753923300484, 9762043084514704, 63858040015802256

Offset: 0

Christoph Koutschan, Feb 27 2025

nonn

			Here are 5 out of 16 solutions for n = 2:
  0 0 1 1   0 0 1 1   0 0 1 1   0 0 1 1   0 1 1 0
  0 0 1 1   0 1 1 0   1 0 0 1   1 1 0 0   1 0 0 1
  1 1 0 0   1 1 0 0   1 1 0 0   1 1 0 0   1 0 0 1
  1 1 0 0   1 0 0 1   0 1 1 0   0 0 1 1   0 1 1 0
the remaining ones are obtained from these by reflecting, rotating, or exchanging 0 and 1.

Christoph Koutschan, Table of n, a(n) for n = 0..1000
Robert Dougherty-Bliss, Christoph Koutschan, Natalya Ter-Saakov, and Doron Zeilberger, The (Symbolic and Numeric) Computational Challenges of Counting 0-1 Balanced Matrices, Enumerative Combinatorics and Applications 5:2, Article #S2R14, 2025.
Christoph Koutschan, Recurrence of order 10 with polynomial coefficients of degree 21.

Cf. A381553, A381554.

a(n) ~ 4*phi^(4*n)/(Pi*sqrt(5)*n), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Feb 27 2025

A306420 Maximal Laman number among all minimally rigid graphs on n vertices.

Original entry on oeis.org

1, 1, 2, 4, 8, 24, 56, 136, 344, 880, 2288, 6180, 15536, 42780

Offset: 1

Christoph Koutschan, Feb 14 2019

The Laman number gives the number of (complex) embeddings of a minimally rigid graph in 2D, modulo translations and rotations, when the edge lengths of the graph are chosen generically. In general, this number is larger than the number of real embeddings. Equivalently, the Laman number of a graph is the number of complex solutions of the quadratic polynomial system {x_1 = y_1 = x_2 = 0, y_2 = l(1,2), (x_i - x_j)^2 + (y_i - y_j)^2 = l(i,j)^2}, for all (i,j) such that the vertices i and j are connected by an edge (w.l.o.g. we assume that there is an edge between the vertices 1 and 2). The quantities l(i,j) correspond to the prescribed edge "lengths" (they can also be complex numbers).
A graph that is constructed only by Henneberg moves of type 1 (i.e., adding one new vertex and connecting it with two existing vertices), has Laman number 2^(n-2). The smallest minimally rigid graph that cannot be constructed in this way, is the 3-prism graph with 6 vertices. Therefore the sequence grows faster than 2^(n-2) for n >= 6.
We know that a graph with n <= 13 vertices achieving the maximal Laman number is unique. We do not know if this is necessarily true for more vertices.

			A graph with one vertex can be drawn in the plane in a unique way, and similarly the graph with two vertices connected by an edge. The unique minimally rigid graph with three vertices is the triangle, which admits two different embeddings (they differ by reflection). The unique minimally rigid graph with four vertices is a quadrilateral with one diagonal (i.e., we have five edges). By fixing the diagonal, each of the two triangles can be flipped independently, yielding four different embeddings.

J. Capco, M. Gallet, G. Grasegger, C. Koutschan, N. Lubbes, J. Schicho, The number of realizations of a Laman graph, SIAM Journal on Applied Algebra and Geometry 2(1), pp. 94-125, 2018.
I. Z. Emiris, E. P. Tsigaridas, A. E. Varvitsiotis, Algebraic methods for counting Euclidean embeddings of graphs. Graph Drawing: 17th International Symposium, pp. 195-200, 2009.
G. Grasegger, C. Koutschan, E. Tsigaridas, Lower bounds on the number of realizations of rigid graphs, Experimental Mathematics, 2018 (doi: 10.1080/10586458.2018.1437851).

Jose Capco, Nauty plugin to compute maximal Laman numbers.
Jose Capco, Matteo Gallet, Georg Grasegger, Christoph Koutschan, Niels Lubbes, and Josef Schicho, The number of realizations of a Laman graph (website).
Jose Capco, Matteo Gallet, Georg Grasegger, Christoph Koutschan, Niels Lubbes, and Josef Schicho, The number of realizations of a Laman graph, arXiv:1701.05500 [math.AG], 2017
Ioannis Z. Emiris and Guillaume Moroz, The assembly modes of rigid 11-bar linkages, IFToMM 2011 World Congress, Guanajuato, Mexico, 2011; arXiv:1010.6214 [cs.RO], 2010-2017.
Georg Grasegger, Explorations on the number of realizations of minimally rigid graphs, arXiv:2502.04736 [math.CO], 2025. See p. 24.
Georg Grasegger, Christoph Koutschan, and Elias Tsigaridas, Lower bounds on the number of realizations of rigid graphs, arXiv:1710.08237 [math.CO], 2017-2018.
Christoph Koutschan and Jose Capco, Latest C++ implementation [Broken link]
G. Laman, On Graphs and Rigidity of Planar Skeletal Structures, J. Engineering Mathematics, Vol. 4, No. 4, 1970, pp. 331-340.
H. Pollaczek-Geiringer, Über die Gliederung ebener Fachwerke, Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 7, No. 1, 1927, pp. 58-72.
Wikipedia, Laman graph

Cf. A227117, A273468.

nauty
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a(13) computed by Jose Capco added by Christoph Koutschan, Feb 15 2019

a(14) computed and added by Jose Capco, Oct 02 2023

A273468 Number of minimally rigid graphs with n vertices constructible by Henneberg type I moves.

Original entry on oeis.org

1, 1, 1, 1, 3, 11, 61, 499, 5500, 75635, 1237670, 23352425, 498028767, 11836515526, 310152665647, 8883427573134

Offset: 1

Christoph Koutschan, May 23 2016

A graph is called rigid if, when we fix the length of each edge, it has only finitely many embeddings in the plane. A graph is called minimally rigid (or a Laman graph) if there is no edge that can be omitted while keeping the rigidity property. Laman graphs can be constructed by applying successively Henneberg moves (of type I or type II), starting with the graph that consists of two vertices joined by an edge. Here we consider Laman graphs that can be obtained by using only Henneberg type I moves, which means: adding one vertex and joining it with two different existing vertices.

			A single vertex is rigid.
The graph consisting of two vertices joined by an edge is rigid.
A triangle is rigid and it is obtained by a single Henneberg type I move.
One more such move yields the only Laman graph with four vertices.
Also all three Laman graphs with five vertices can be constructed with type I moves. Therefore the first five entries of this sequence agree with A227117.
An example of a Laman graph that cannot be constructed using only Henneberg type I moves is the complete bipartite graph K(3,3).

L. Henneberg, Die graphische Statik der starren Systeme, Leipzig, 1911.
Christoph Koutschan, Mathematica program
G. Laman, On Graphs and Rigidity of Plane Skeletal Structures, Journal of Engineering Mathematics 4 (1970), 331-340.
Martin Larsson, Nauty Laman plugin
Wikipedia, Laman graph

Cf. A227117.

Table[Length[H1LamanGraphs[n]], {n,3,7}]  (* see link *)

gensparseg $n -H -u # With Laman plugin; see link.

a(13) added by Jose Capco, Dec 07 2018
a(14)-a(15) added by Martin Larsson, Dec 21 2020
a(16) from Martin Larsson added by Max Alekseyev, Jan 14 2025

A271671 Number of n-step excursions on the 8-dimensional f.c.c. lattice.

Original entry on oeis.org

1, 0, 112, 2688, 126000, 6316800, 364887040, 23038364160, 1562288430640, 112014905049600, 8399872737107712, 653454438359331840, 52412319029000899584, 4313870772211888183296, 362994066330649023029760

Offset: 0

Christoph Koutschan, Apr 12 2016

a(n) = number of walks in the integer lattice Z^8 starting and ending at the origin, using only the steps of the form (s_1, ..., s_8) with s_1^2 + ... + s_8^2 = 2, i.e., each possible step has precisely two nonzero entries which can be +1 or -1.

			There is one walk with no steps.
No walk with a single steps returns to the origin.
The number of returning walks with two steps is exactly the number of allowed steps (called the coordination number of the lattice): a(2) = 4*binomial(8,2).

Christoph Koutschan, Table of n, a(n) for n = 0..493
S. Hassani, C. Koutschan, J-M. Maillard, N. Zenine, Lattice Green Functions: the d-dimensional face-centred cubic lattice, d = 8, 9, 10, 11, 12, arXiv:1601.05657 [math-ph], 2016.
S. Hassani, C. Koutschan, J-M. Maillard, N. Zenine, Lattice Green functions: the d-dimensional face-centred cubic lattice, d = 8, 9, 10, 11, 12, Journal of Physics A: Mathematical and Theoretical 49(16) (2016), 164003.
C. Koutschan, Computations for higher-dimensional fcc lattices.
C. Koutschan, Differential operator annihilating the generating function.
C. Koutschan, Recurrence equation.

Cf. A002899 (d = 3, i.e., excursions on the 3-dimensional f.c.c. lattice), A271432 (d = 4), A271650 (d = 5), A271651 (d = 6), A271670 (d = 7), this sequence (d = 8), A271672 (d = 9), A271673 (d = 10), A271674 (d = 11).

nmax := 50: tt := [seq([seq(add(binomial(2*p,p)*binomial(2*j,2*p-n)*binomial(2*n+2*j-2*p,n+j-p), p = floor((n+1)/2)..floor((n+2*j)/2)), j = 0..floor((nmax-n)/2))], n = 0..nmax)]: for d1 from 3 to 8 do tt := [seq([seq(add(binomial(n,p)*add(binomial(2*j,2*q-p)*binomial(2*j+2*p-2*q,j+p-q)*tt[n-p+1,q+1], q = floor((p+1)/2)..floor((p+2*j)/2)), p = 0..n), j = 0..floor((nmax-n)/2))], n = 0..nmax)]: od: [seq(tt[n+1,1], n = 0..nmax)];

nmax = 50; T = Table[Sum[Binomial[2 p, p]*Binomial[2 j, 2 p - n]*Binomial[2 n + 2 j - 2 p, n + j - p], {p, Floor[(n + 1)/2], Floor[(n + 2 j)/2]}], {n, 0, nmax}, {j, 0, Floor[(nmax - n)/2]}]; Do[T = Table[Sum[Binomial[n, p]*Sum[Binomial[2 j, 2 q - p]*Binomial[2 j + 2 p - 2 q, j + p - q]*T[[n - p + 1, q + 1]], {q, Floor[(p + 1)/2], Floor[(p + 2 j)/2]}], {p, 0, n}], {n, 0, nmax}, {j, 0, If[d1 < 8, Floor[(nmax - n)/2], 0]}], {d1, 3, 8}]; First /@ T

a(n) conjecturally satisfies a linear recurrence equation of order 20 with polynomial coefficients of degree 109 (see link above).

The probability generating function P(z) = Sum_{n>=0} a(n)*(z/112)^n is given by the 8-fold integral (1/Pi)^8 Int_{0..Pi} ... Int_{0..Pi} 1/(1-z*lambda_8) dk_1 ... dk_8, where the structure function is defined as lambda_8 = (1/binomial(8,2)) Sum_{i=1..8} Sum_{j=(i+1)..8} cos(k_i)*cos(k_j). The function P(z) conjecturally satisfies a linear ODE of order 14 with polynomial coefficients of degree 126 (see link above).

A271670 Number of n-step excursions on the 7-dimensional f.c.c. lattice.

Original entry on oeis.org

1, 0, 84, 1680, 66276, 2731680, 128704800, 6555265920, 355588928100, 20247799145280, 1198746727590384, 73266532153214400, 4598338364703822816, 295145004688715301120, 19311431876483926443264

Offset: 0

Christoph Koutschan, Apr 12 2016

a(n) = number of walks in the integer lattice Z^7 starting and ending at the origin, using only the steps of the form (s_1, ..., s_7) with s_1^2 + ... + s_7^2 = 2, i.e., each possible step has precisely two nonzero entries which can be +1 or -1.

			There is one walk with no steps.
No walk with a single steps returns to the origin.
The number of returning walks with two steps is exactly the number of allowed steps (called the coordination number of the lattice): a(2) = 4*binomial(7,2).

Christoph Koutschan, Table of n, a(n) for n = 0..524
C. Koutschan, Computations for higher-dimensional fcc lattices.
C. Koutschan, Differential operator annihilating the generating function.
C. Koutschan, Recurrence equation.
N. Zenine, S. Hassani, J-M. Maillard, Lattice Green Functions: the seven-dimensional face-centred cubic lattice, arXiv:1409.8615 [math-ph], 2014.
N. Zenine, S. Hassani, J-M. Maillard, Lattice Green Functions: the seven-dimensional face-centred cubic lattice, Journal of Physics A: Mathematical and Theoretical 48 (2015), 035205.

Cf. A002899 (d = 3, i.e., excursions on the 3-dimensional f.c.c. lattice), A271432 (d = 4), A271650 (d = 5), A271651 (d = 6), this sequence (d = 7), A271671 (d = 8), A271672 (d = 9), A271673 (d = 10), A271674 (d = 11).

nmax := 50: tt := [seq([seq(add(binomial(2*p,p)*binomial(2*j,2*p-n)*binomial(2*n+2*j-2*p,n+j-p), p = floor((n+1)/2)..floor((n+2*j)/2)), j = 0..floor((nmax-n)/2))], n = 0..nmax)]: for d1 from 3 to 7 do tt := [seq([seq(add(binomial(n,p)*add(binomial(2*j,2*q-p)*binomial(2*j+2*p-2*q,j+p-q)*tt[n-p+1,q+1], q = floor((p+1)/2)..floor((p+2*j)/2)), p = 0..n), j = 0..floor((nmax-n)/2))], n = 0..nmax)]: od: [seq(tt[n+1,1], n = 0..nmax)];

nmax = 50; T = Table[Sum[Binomial[2 p, p]*Binomial[2 j, 2 p - n]*Binomial[2 n + 2 j - 2 p, n + j - p], {p, Floor[(n + 1)/2], Floor[(n + 2 j)/2]}], {n, 0, nmax}, {j, 0, Floor[(nmax - n)/2]}]; Do[T = Table[Sum[Binomial[n, p]*Sum[Binomial[2 j, 2 q - p]*Binomial[2 j + 2 p - 2 q, j + p - q]*T[[n - p + 1, q + 1]], {q, Floor[(p + 1)/2], Floor[(p + 2 j)/2]}], {p, 0, n}], {n, 0, nmax}, {j, 0, If[d1 < 7, Floor[(nmax - n)/2], 0]}], {d1, 3, 7}]; First /@ T

a(n) conjecturally satisfies a linear recurrence equation of order 15 with polynomial coefficients of degree 56 (see link above).

The probability generating function P(z) = Sum_{n>=0} a(n)*(z/84)^n is given by the 7-fold integral (1/Pi)^7 Int_{0..Pi} ... Int_{0..Pi} 1/(1-z*lambda_7) dk_1 ... dk_7, where the structure function is defined as lambda_7 = (1/binomial(7,2)) Sum_{i=1..7} Sum_{j=(i+1)..7} cos(k_i)*cos(k_j). The function P(z) conjecturally satisfies an eleventh-order linear ODE with polynomial coefficients of degree 68 (see link above).

A271673 Number of n-step excursions on the 10-dimensional f.c.c. lattice.

Original entry on oeis.org

1, 0, 180, 5760, 355860, 24226560, 1923670800, 169658496000, 16291413249300, 1674631754611200, 181989927592033680, 20709782925396364800, 2449425950787336166800, 299337868552812779289600, 37621311095831818078152000

Offset: 0

Christoph Koutschan, Apr 12 2016

a(n) = number of walks in the integer lattice Z^10 starting and ending at the origin, using only the steps of the form (s_1, ..., s_10) with s_1^2 + ... + s_10^2 = 2, i.e., each possible step has precisely two nonzero entries which can be +1 or -1.

			There is one walk with no steps.
No walk with a single steps returns to the origin.
The number of returning walks with two steps is exactly the number of allowed steps (called the coordination number of the lattice): a(2) = 4*binomial(10,2).

Christoph Koutschan, Table of n, a(n) for n = 0..449
S. Hassani, C. Koutschan, J-M. Maillard, N. Zenine, Lattice Green Functions: the d-dimensional face-centred cubic lattice, d = 8, 9, 10, 11, 12, arXiv:1601.05657 [math-ph], 2016.
S. Hassani, C. Koutschan, J-M. Maillard, N. Zenine, Lattice Green functions: the d-dimensional face-centred cubic lattice, d = 8, 9, 10, 11, 12, Journal of Physics A: Mathematical and Theoretical 49(16) (2016), 164003.
C. Koutschan, Computations for higher-dimensional fcc lattices.
C. Koutschan, Differential operator annihilating the generating function.
C. Koutschan, Recurrence equation.

Cf. A002895, A002899 (d = 3, i.e., excursions on the 3-dimensional f.c.c. lattice), A271432 (d = 4), A271650 (d = 5), A271651 (d = 6), A271670 (d = 7), A271671 (d = 8), A271672 (d = 9), this sequence (d = 10), A271674 (d = 11).

nmax := 50: tt := [seq([seq(add(binomial(2*p,p)*binomial(2*j,2*p-n)*binomial(2*n+2*j-2*p,n+j-p), p = floor((n+1)/2)..floor((n+2*j)/2)), j = 0..floor((nmax-n)/2))], n = 0..nmax)]: for d1 from 3 to 10 do tt := [seq([seq(add(binomial(n,p)*add(binomial(2*j,2*q-p)*binomial(2*j+2*p-2*q,j+p-q)*tt[n-p+1,q+1], q = floor((p+1)/2)..floor((p+2*j)/2)), p = 0..n), j = 0..floor((nmax-n)/2))], n = 0..nmax)]: od: [seq(tt[n+1,1], n = 0..nmax)];

nmax = 50; T = Table[Sum[Binomial[2 p, p]*Binomial[2 j, 2 p - n]*Binomial[2 n + 2 j - 2 p, n + j - p], {p, Floor[(n + 1)/2], Floor[(n + 2 j)/2]}], {n, 0, nmax}, {j, 0, Floor[(nmax - n)/2]}]; Do[T = Table[Sum[Binomial[n, p]*Sum[Binomial[2 j, 2 q - p]*Binomial[2 j + 2 p - 2 q, j + p - q]*T[[n - p + 1, q + 1]], {q, Floor[(p + 1)/2], Floor[(p + 2 j)/2]}], {p, 0, n}], {n, 0, nmax}, {j, 0, If[d1 < 10, Floor[(nmax - n)/2], 0]}], {d1, 3, 10}]; First /@ T

a(n) conjecturally satisfies a linear recurrence equation of order 30 with polynomial coefficients of degree 274 (see link above).

The probability generating function P(z) = Sum_{n>=0} a(n)*(z/180)^n is given by the 10-fold integral (1/Pi)^10 Int_{0..Pi} ... Int_{0..Pi} 1/(1-z*lambda_10) dk_1 ... dk_10, where the structure function is defined as lambda_10 = (1/binomial(10,2)) Sum_{i=1..10} Sum_{j=(i+1)..10} cos(k_i)*cos(k_j). The function P(z) conjecturally satisfies a linear ODE of order 22 with polynomial coefficients of degree 300 (see link above).

A271672 Number of n-step excursions on the 9-dimensional f.c.c. lattice.

Original entry on oeis.org

1, 0, 144, 4032, 219024, 12942720, 887135040, 67057079040, 5484251057040, 477369708721920, 43704143706754944, 4170816570389736960, 412062922497680790336, 41920366214226928716288, 4372905161028532447478016

Offset: 0

Christoph Koutschan, Apr 12 2016

a(n) = number of walks in the integer lattice Z^9 starting and ending at the origin, using only the steps of the form (s_1, ..., s_9) with s_1^2 + ... + s_9^2 = 2, i.e., each possible step has precisely two nonzero entries which can be +1 or -1.

			There is one walk with no steps.
No walk with a single steps returns to the origin.
The number of returning walks with two steps is exactly the number of allowed steps (called the coordination number of the lattice): a(2) = 4*binomial(9,2).

Christoph Koutschan, Table of n, a(n) for n = 0..469
S. Hassani, C. Koutschan, J-M. Maillard, N. Zenine, Lattice Green Functions: the d-dimensional face-centred cubic lattice, d = 8, 9, 10, 11, 12, arXiv:1601.05657 [math-ph], 2016.
S. Hassani, C. Koutschan, J-M. Maillard, N. Zenine, Lattice Green functions: the d-dimensional face-centred cubic lattice, d = 8, 9, 10, 11, 12, Journal of Physics A: Mathematical and Theoretical 49(16) (2016), 164003.
C. Koutschan, Computations for higher-dimensional fcc lattices.
C. Koutschan, Differential operator annihilating the generating function.
C. Koutschan, Recurrence equation.

Cf. A002899 (d = 3, i.e., excursions on the 3-dimensional f.c.c. lattice), A271432 (d = 4), A271650 (d = 5), A271651 (d = 6), A271670 (d = 7), A271671 (d = 8), this sequence (d = 9), A271673 (d = 10), A271674 (d = 11).

nmax := 50: tt := [seq([seq(add(binomial(2*p,p)*binomial(2*j,2*p-n)*binomial(2*n+2*j-2*p,n+j-p), p = floor((n+1)/2)..floor((n+2*j)/2)), j = 0..floor((nmax-n)/2))], n = 0..nmax)]: for d1 from 3 to 9 do tt := [seq([seq(add(binomial(n,p)*add(binomial(2*j,2*q-p)*binomial(2*j+2*p-2*q,j+p-q)*tt[n-p+1,q+1], q = floor((p+1)/2)..floor((p+2*j)/2)), p = 0..n), j = 0..floor((nmax-n)/2))], n = 0..nmax)]: od: [seq(tt[n+1,1], n = 0..nmax)];

nmax = 50; T = Table[Sum[Binomial[2 p, p]*Binomial[2 j, 2 p - n]*Binomial[2 n + 2 j - 2 p, n + j - p], {p, Floor[(n + 1)/2], Floor[(n + 2 j)/2]}], {n, 0, nmax}, {j, 0, Floor[(nmax - n)/2]}]; Do[T = Table[Sum[Binomial[n, p]*Sum[Binomial[2 j, 2 q - p]*Binomial[2 j + 2 p - 2 q, j + p - q]*T[[n - p + 1, q + 1]], {q, Floor[(p + 1)/2], Floor[(p + 2 j)/2]}], {p, 0, n}], {n, 0, nmax}, {j, 0, If[d1 < 9, Floor[(nmax - n)/2], 0]}], {d1, 3, 9}]; First /@ T

a(n) conjecturally satisfies a linear recurrence equation of order 22 with polynomial coefficients of degree 151 (see link above).
The probability generating function P(z) = Sum_{n>=0} a(n)*(z/144)^n is given by the 9-fold integral (1/Pi)^9 Int_{0..Pi} ... Int_{0..Pi} 1/(1-z*lambda_9) dk_1 ... dk_9, where the structure function is defined as lambda_9 = (1/binomial(9,2)) Sum_{i=1..9} Sum_{j=(i+1)..9} cos(k_i)*cos(k_j). The function P(z) conjecturally satisfies a linear ODE of order 18 with polynomial coefficients of degree 169 (see link above).
Hence a(n) conjecturally satisfies a linear recurrence equation with polynomial coefficients.

A271674 Number of n-step excursions on the 11-dimensional f.c.c. lattice.

Original entry on oeis.org

1, 0, 220, 7920, 548460, 42276960, 3818372800, 385303564800, 42556023409900, 5056698223684800, 638162986199119920, 84683717201322993600, 11723112517163129913600, 1682392957299926013542400, 249030549709148521993536000, 37864267170542400351711467520

Offset: 0

Christoph Koutschan, Apr 12 2016

a(n) = number of walks in the integer lattice Z^11 starting and ending at the origin, using only the steps of the form (s_1, ..., s_11) with s_1^2 + ... + s_11^2 = 2, i.e., each possible step has precisely two nonzero entries which can be +1 or -1.

			There is one walk with no steps.
No walk with a single steps returns to the origin.
The number of returning walks with two steps is exactly the number of allowed steps (called the coordination number of the lattice): a(2) = 4*binomial(11,2).

Christoph Koutschan, Table of n, a(n) for n = 0..433
S. Hassani, C. Koutschan, J-M. Maillard, N. Zenine, Lattice Green Functions: the d-dimensional face-centred cubic lattice, d = 8, 9, 10, 11, 12, arXiv:1601.05657 [math-ph], 2016.
S. Hassani, C. Koutschan, J-M. Maillard, N. Zenine, Lattice Green functions: the d-dimensional face-centred cubic lattice, d = 8, 9, 10, 11, 12, Journal of Physics A: Mathematical and Theoretical 49(16) (2016), 164003.
C. Koutschan, Computations for higher-dimensional fcc lattices.
C. Koutschan, Differential operator annihilating the generating function.

Cf. A002899 (d = 3, i.e., excursions on the 3-dimensional f.c.c. lattice), A271432 (d = 4), A271650 (d = 5), A271651 (d = 6), A271670 (d = 7), A271671 (d = 8), A271672 (d = 9), A271673 (d = 10), this sequence (d = 11).

nmax := 50: tt := [seq([seq(add(binomial(2*p,p)*binomial(2*j,2*p-n)*binomial(2*n+2*j-2*p,n+j-p), p = floor((n+1)/2)..floor((n+2*j)/2)), j = 0..floor((nmax-n)/2))], n = 0..nmax)]: for d1 from 3 to 11 do tt := [seq([seq(add(binomial(n,p)*add(binomial(2*j,2*q-p)*binomial(2*j+2*p-2*q,j+p-q)*tt[n-p+1,q+1], q = floor((p+1)/2)..floor((p+2*j)/2)), p = 0..n), j = 0..floor((nmax-n)/2))], n = 0..nmax)]: od: [seq(tt[n+1,1], n = 0..nmax)];

nmax = 50; T = Table[Sum[Binomial[2 p, p]*Binomial[2 j, 2 p - n]*Binomial[2 n + 2 j - 2 p, n + j - p], {p, Floor[(n + 1)/2], Floor[(n + 2 j)/2]}], {n, 0, nmax}, {j, 0, Floor[(nmax - n)/2]}]; Do[T = Table[Sum[Binomial[n, p]*Sum[Binomial[2 j, 2 q - p]*Binomial[2 j + 2 p - 2 q, j + p - q]*T[[n - p + 1, q + 1]], {q, Floor[(p + 1)/2], Floor[(p + 2 j)/2]}], {p, 0, n}], {n, 0, nmax}, {j, 0, If[d1 < 11, Floor[(nmax - n)/2], 0]}], {d1, 3, 11}]; First /@ T

The probability generating function P(z) = Sum_{n>=0} a(n)*(z/220)^n is given by the 11-fold integral (1/Pi)^11 Int_{0..Pi} ... Int_{0..Pi} 1/(1-z*lambda_11) dk_1 ... dk_11, where the structure function is defined as lambda_11 = (1/binomial(11,2)) Sum_{i=1..11} Sum_{j=(i+1)..11} cos(k_i)*cos(k_j). The function P(z) conjecturally satisfies a linear ODE of order 27 with polynomial coefficients of degree 409 (see link above).

Hence a(n) conjecturally satisfies a linear recurrence equation with polynomial coefficients.

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User: Christoph Koutschan

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